As \(P\)  is constant on the \(\times \) -orbits, \(\mathrm {d}P\)  has rank at most \(n-1\) . On the other hand, the term \((\lambda (A)I-A)\mathrm {d}v\)  has the same rank as \(\lambda (A)I-A\) , which is \(n-1\)  by non-degeneracy. Hence \(\mathrm {d}P\)  has constant rank \(n-1\) , and so by the Constant Rank Theorem [1]} the fibers of \(P\)  are closed submanifolds of (complex) dimension \((n^2+n)-(n-1)=n^2+1\) . The maps \(\pi _{\lambda v}\)  and \(\pi _v\)  are then restrictions of smooth projections to a submanifold, hence smooth.
